Questions tagged [topology]

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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51 views

Why there are only two manifold in 1d?

I'm following Witten's essay and he writes: Let us try to make such a theory with one spacetime dimension instead of four. The choices for a one-manifold are quite limited: and then gives this ...
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After conformal compactification, do null geodesics intersect future null infinity nonasymptotically?

I was going through this paper and was worried about an assumption in the main proof (Theorem 3.1), where they assume null geodesics intersect the boundary extension after conformal compactification ...
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What does “continuous transformation” mean with regard to the Hamiltonian of a system?

When dealing with topological phases of matter (topological insulators, quantum hall effect, etc...) one says that the system remains in the same phase as long as any continuous transformation of the ...
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Closed space curve

Given a smooth space curve, it can be uniquely determined by the curvature $\kappa$ and torsion $\tau$ (Frenet–Serret assuming the curvature doen't vanish). Assuming the curve is closed and has ...
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34 views

What is the motivation for the $\mathrm{Spin}(n)$ group to be the double cover of $SO(n)$? [duplicate]

The $\mathrm{Spin}(n)$ group is defined to be the double cover of $\mathrm{SO}(n)$. In the case of $n > 2$, this agrees with the universal cover. However, for $n=2$, the physically relevant group ...
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What is physical significance of $k$-space topology?

I am working with topological materials and I do understand what topology is. While reading papers, I am not able to understand the "nontrivial k-space topology". How skyrmions are mostly ...
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Why use the double cover instead of the universal cover of $SO(2)$?

In QM, we are generally looking for projective representations of the underlying symmetry group (because of Wigner's theorem). So generally, we should be looking for representations of central ...
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46 views

Effect of global topology of space on wavefunctions?

Usually, we have solve for wavefunctions assuming trivial periodic boundary conditions i.e. we connect, in 2D for example, like a torus. What would be the effect on the spectra or eigenfunctions of a ...
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The theta term and triviality of principal bundles

Apologies if this question is trivial or has been answered before. If we consider a Yang-Mills theory (with a simple, compact Lie group $G$) on $\mathbb{R}^4$, it is well-known that all the finite-...
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Difference between quantum anomalous Hall (QAH) insulator and Chern insulator?

In the literature, I see QAH insulator defined as a quantum state with quantized Hall response without an external magnetic field. Time-reversal symmetry is instead broken by different means such as ...
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Rigorous definition of “condensation” in Topological Lattice Gauge theories?

I have encountered the term "string-net condensation" in the string-net paper by Levin-Wen, and in Kitaev's Toric Code - where we say that the ground state is a string net condensate and (I ...
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What are all the types of topologically-relevant band degeneracies in contemporary 2D condensed matter research? What about those often ignored?

Cross-posted the question here: https://mattermodeling.stackexchange.com/q/4877/1766 In the study of 2D condensed matter systems, I have seen several kinds of band degeneracies. I call 'bands' the ...
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Cosmology - Confusion About Visualising the Universe as the Surface of a 3-Sphere

Consider the FRW metric for the Universe in the form found in many standard cosmology textbooks: $$ds^2 = -dt^2 + a(t)^2\left(\frac{dr^2}{1-Kr^2}+r^2(d\theta^2 + \sin^2\theta d\phi^2)\right)$$ I am ...
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Topology of spacetime

We know that spacetime is an orientable manifold: Can spacetime be non-orientable? But supposing that spacetime is an orientable closed 2D surface, one might envision a variety of non-equivalent ...
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Interpretation of black hole metric with fractional $\kappa$ instead of the usual $\kappa\in\{-1,0,1\}$

The metric for a black hole can be written: $$d s^{2}=-\left(\kappa-\frac{2 M}{r}\right) d t^{2}+\left(\kappa-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2} d \Sigma_{2, k}^{2}$$ where $\kappa=-1,0,1$ ...
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Could torus spaces (defined by Planck length) exist?

Let $\epsilon_c$ denote critical energy density, $\epsilon_t$ energy mass density at time $t$, $c$ the speed of light, $G$ the gravitational constant, $H=\dot a/ a$ the Hubble constant, $\Lambda$ the ...
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Can you conformally map the Lorentzian cylinder to the Lorentzian plane?

When studying CFT in Euclidean signature, for the purpose of radial quantization, we conformally map the Euclidean cylinder to the Euclidean plane (minus the origin, which I ignore). Can one also ...
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Topological meaning of the integral of the trace of Cartan-Maurer forms in Anthony Zee's book on QFT in a nutshell

I learnt in S. Sternberg's book "Curvature in Mathematics & Physics" over the Maurer-Cartan form that if there is a tangent vector $v \in TG_g$ at a point $g \in G$ ($G$ is supposed to ...
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How the wave vector $k$ change slowly and travel a loop in Brillouin zone when we calculate the Berry phase?

According to the definition of the Berry phase, there must have a slowly changing parameter that travel a loop. when we discuss topology in energy band, the slowly changing parameter seems the $k$. my ...
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1answer
67 views

Topology - The space of time [duplicate]

In physics, one of the ways to solve problems is to find a similar mathematical model which describes the problem, and that leads to my question: By experiments we familiar with the space of time, we ...
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2answers
190 views

Topology in cosmology

Usually in cosmology, we make the hypothesis that the universe is isotropic. Which conditions does this hypothesis impose on the topology of the universe? Does it fix completely the topology? Are all ...
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28 views

Chinese finger-trap and other nets which assume different 3D shapes under different positions and loads

Please allow a question for general orientation of a newbie lay-person. I just came from a different stack exchange site seeing with curiosity a link to this question: Is the surface of a sphere and a ...
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506 views

What is Eric Weinstein's Geometric Unity theory? [closed]

I can usually follow the basic ideas of a theory, but Weinstein's Geometric Unity theory is completely incomprehensible to me. It leads me to suspect that it is high level crackpottery, but he seems ...
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Geometric features of a closed finite universe

I am a student, so the question may sound silly. If the 2-sphere is the surface of a ball, that is, it is embedded in a three-dimensional space, then the 3-sphere must also be the surface of a four-...
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39 views

Discriminate closed-chain from open-chain

I have two classes of chains: closed-chains where a random path ends near where it starts (ie. loop), and open-chains without this restriction (ie. random walk). These chains are a directed graph ...
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83 views

Transitive Closure of Spacelike Separation

Let $S$ be a set of (possibly infinitely many) events in Minkowski spacetime. What would be the necessary and sufficient condition for $S$ (or the elements of S) to be such that for any $x, y, z$ $\in ...
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1answer
103 views

Is space-like separation transitive?

Suppose that events $A$ and $B$ are spacelike separated. Also suppose that events $B$ and $C$ are spacelike separated. Does this guarantee that $A$ and $C$ are spacelike separated? That is, is the ...
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141 views

Explain why the universe could be compact

Regarding the topology of the universe, it could be compact like a sphere or open like a Euclidean space, but since the universe started from a single point, doesn't that mean that the shape of the ...
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1answer
156 views

Time-reversal symmetry for spin Hamiltonian

In the topology online course by TU Delft, the time-reversal operator acting on a system of spin-1/2 particles is introduced as $$ \mathcal T = i\sigma_y\mathcal K. $$ I understand this acts on the ...
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61 views

An instanton in $d$ dimensions is often a soliton in $d + 1$ dimensions?

The title of this questions is a "folklore" I've heard from a lot of researchers, but I never understood why this is the case. I know what an instanton and soliton is, respectively in the ...
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Are topological changes to dynamic spacetime quantized? Can the Chern-Gauss-Bonnett theorem illuminate dynamics?

I was looking at the Chern-Gauss-Bonnett theorem in dimension 4. Here we can write the Euler characteristic of a compact 4-manifold as: $$\chi(M)=\frac{1}{32\pi^{2}}\intop_{M}\left(|\mathrm{Riem}|^{2}-...
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Spin connection for a paralellization takes more general forms than $SO(3,1)$ in different spacetime topologies?

I'm interested in a frame bundle over spacetimes with different topologies. In the trivial case of Minkowskian space ($\mathbb{R}^{3,1}$), a frame (or tangent space) at one point is going to be ...
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111 views

Why does a choice of this $\psi$ in the worldsheet metric corresponds to a choice of complex structure?

As far as I'm aware, a complex manifold $M$ is a topological manifold together with an atlas ${\cal A}$ of charts $(U_i,\varphi_i)\in{\cal A}$ such that the open sets $U_i$ cover $M$, the maps $\...
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4answers
104 views

Do objects in a 2D universe have an edge?

When discussing a 2D universe, many assume that an object would only be seen as "a line". This would imply that you are seeing the "edge" of the object. But, if there are only ...
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Isometric embedding of embedding of Schwarzchild metric [duplicate]

I am reading through this article https://arxiv.org/abs/1010.4256 about the special case of the positive mass theorem in general relativity. I do not understand the section below: In particular what ...
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898 views

Why the work done in a conservative field around a closed circle does not vanish when calculated in cylindrical coordinates?

I was solving problem 2.4.13 from the book "George B Arfken, Hans J Weber - Mathematical Methods For Physicists- Sixth edition" and the problems was that: Problem 2.4.13 A force is ...
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55 views

Isometric embedding of Schwarzchild metric in $\mathbb{R}^4$

I am reading through this article https://arxiv.org/abs/1010.4256 about the special case of the positive mass theorem in general relativity. I do not understand the section below: In particular what ...
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25 views

Trivial examples for the Chern number from the potential for quantized transport

I'm trying to understand the phenomena of quantized electron transport better. The difficult step is that for any Hamiltonian (where $V(x,t)$ is periodic in both arguments and is a slow function of $t$...
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51 views

Magnetic monopole, vector-potential and differential forms

When written in the language of exterior algebra, Maxwell-Thomson equation writes as $dB=0$ where $d$ is the exterior derivative and $B$ is the magnetic flux 2-form. From Poincaré's lemma, it follows ...
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The volume enclosing the charge in Gauss's law: does it have to be simply connected?

I was trying to apply Gauss's law to a simple problem: Find the capacitance of a cylindrical capacitor. Inner radius is $a$ and outer radius is $c$. The space between the plates is a dielectric ...
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Connectivity in phase-space

In my statistical mechanics lecture, it was claimed that a volume of phase-space cannot be split into two separate volumes as time evolves. I suspect that this is a topological fact that I am not ...
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Is there physical reason for a stably causal spacetime, or the existence of a Cauchy surface?

In their 1979 essay Global structure of spacetimes, Geroch and Horowitz describe methods of determining the topology, causal structure and singularity of spacetimes. Their (mathematical) arguments are ...
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1answer
63 views

Is it enough to give a time-orientation to define a spin structure?

Maybe I got it wrong and my question doesn't make sense, excuse me if that's the case. For a smooth Lorentz 4-manifold $(M, g)$ with signature $(- + + +)$ is it enough to give a time-orientation to ...
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lecture notes about the relation between algebraic topology, topological quantum field theory, condensed matter physics [closed]

I am an undergraduate student and I am very interested in topology with its application in physics. So last year I've read some books about this field, mainly about topological soliton, some ...
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224 views

Can magnetic loops with no source current knot or link?

The answer to this question is obviously no. I would like to pose a variation of that question. Suppose a simply connected domain of a 3-d vacuum space has no source current. Does there exist a case ...
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Magnetic field loops do not knot or link

The magnetic field is composed of closed loops (assuming there is no magnetic monopole). How does one prove any two magnetic loops do not knot to form a link?
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Does a positive curvature necessarily indicate the finiteness of the universe?

Imagine the following situation: more and more accurate measurements of the average density of the Universe reveal that it is greater than the critical one, which corresponds to the model of a closed ...
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About Chern Insulator

I know when we talk about Insulator, U(1)charge symmetry naturally exists. But in this article:Quantum phase transitions of topological insulators without gap closing, the author claims that: "...
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1answer
73 views

Spin 1/2 as belt trick in a smooth field

In the (English) Wikipedia article on Spinor, there is an animation, demonstrating the Dirac belt trick as a model for Spin 1/2. My interpretation of that animation goes like this: If you rotate an ...
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34 views

If the universe had a topological hole, would moving around the hole have a centripetal force?

I understand that if it were of a toroidal topology, it would not literally mean that the universe is in the shape of a 3D donut. However, I can't seem to draw intuition on why or why not it may be ...

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